标题： 环境中有稀有异常的随机游走 显示英文标题

标题： Random Walks in Random Environments with Rare Anomalies

摘要： 我们研究了在$\mathbb{Z}^d$上独立同分布随机环境中的一类随机游走，其中顶点有两种基本类型，我们称之为“蓝色”和“红色”。每种颜色代表了转移概率向量上的不同概率分布。我们介绍了一种研究这些随机游走的方法，该方法通过比较某一特定站点在该站点为红色事件下所花费的期望时间与该站点为蓝色事件下所花费的期望时间，来分析随机游走的行为。这种方法可以得到关于随机游走渐近速度的显式界值。我们重新获得了 Kalikow 的早期结果，但给出了关于速度的新界值。 接下来，我们考虑了一个“稀有异常”模型，在该模型中，绝大多数站点是蓝色的，并且蓝色站点在整个环境中是一致椭圆的，同时在淬火漂移上有一些几乎必然的界值。我们证明了如果红色站点满足在两个固定且非平行方向上的某种一致椭圆性假设，则即使红色站点破坏了淬火漂移的几乎必然界值，只要使红色站点足够罕见，就可以获得随机游走渐近速度的界值，这些界值可以接近蓝色站点处淬火漂移的界值。尤为重要的是，为了实现这一点所需的蓝色站点的比例 $p^*$不依赖于红色站点的分布，除了通过两个方向上的一致椭圆性假设外。 我们的证明基于一种耦合技术，其中两条随机游走运行在环境完全相同的情况下，除了一个顶点之外。当它们到达该顶点时，两条随机游走会解耦，并且我们的证明由重新耦合所需的时间界值驱动。然后，我们通过提供一个反例来展示独立同分布假设的重要性，该反例是在去除此假设后定理陈述的反例。最后，我们以一些开放问题作为结论。 显示英文摘要

摘要： We study random walks in i.i.d. random environments on $\mathbb{Z}^d$ when there are two basic types of vertices, which we call "blue" and "red". Each color represents a different probability distribution on transition probability vectors. We introduce a method of studying these walks that compares the expected amount of time spent at a specific site on the event that the site is red with the expected amount of time spent there on the event that the site is blue. This method produces explicit bounds on the asymptotic velocity of the walk. We recover an early result of Kalikow, but with new bounds on the velocity. Next, we consider a "rare anomaly" model where the vast majority of sites are blue, and blue sites are uniformly elliptic, with some almost-sure bounds on the quenched drift. We show that if the red sites satisfy a certain uniform ellipticity assumption in two fixed, non-parallel directions, then even if red sites break the almost-sure bounds on the quenched drift, making red sites unlikely enough lets us obtain bounds on the asymptotic velocity of the walk arbitrarily close to the bounds on the quenced drift at blue sites. Significantly, the required proportion $p^*$ of blue sites to do this does not depend on the distribution of red sites, except through the uniform ellipticity assumption in two directions. Our proof is based on a coupling technique, where two walks run in environments that are the same everywhere except at one vertex. They decouple when they hit that vertex, and our proof is driven by bounds on how long it takes to recouple. We then demonstrate the importance of the i.i.d. assumption by providing a counterexample to the statement of the theorem with this assumption removed. We conclude with open questions.